Non-Hermitian Rayleigh-Schrödinger perturbation theory

Theoretical Chemistry Accounts, 2016

The partitioning theory provides optical potentials which lead to the resonance energies. The state of the theory in the eighties can be found in the book "Theory of resonances" of Kukulin et al. [3]. The basic principles presented in this book still hold: use of projectors and analytical continuation. Since that time, the theoretical developments have been influenced by the fast development of computers and the theory entered into the quickly expanding field of non-Hermitian quantum mechanics [4]. Jolicard and Austin [5] incorporated optical potentials in computational schemes, and the justification of their method was given by Riss and Meyer [6]. To avoid confusion, we do not use the term optical potential here, but adopt the more suitable expression complex absorbing potential (CAP), as proposed in Ref. [6]. The acronym CAP refers to an energy-independent complex potential added to the Hamiltonian (see review article [7]). For the past two decades, we contributed to the development of approaches combining CAP and perturbation theory [8-12]. In this article, we recognize a resonance as a pole of a Green's function. In Sect. 2, we first recall the definition of the wave function and the energy in the framework of the partitioning technique. Then, a complex absorbing potential written in the form V CAP = −ı ǫ is added to the Hamiltonian. The operator ǫ generalizes the quantity ǫ of collision theory. The parameter in V CAP anticipates that V CAP does not only produce analytic continuation, but that it is also a perturbation operator. The self-energy is expanded in powers of , and the equivalence between this expansion and a Taylor series is demonstrated in Sect. 3 which focuses on our new findings. Finally, the introduction of a convergence operator allows to discuss the convergence properties of these series. Section 4 is devoted to numerical illustrations. First, a discretized N-dimensional Fano model is used to check the accuracy and the convergence properties of the energies. It is shown Abstract We propose a contribution to the theory of quantum resonances that combines complex absorbing potentials (CAP) with standard perturbation theory. We start from resolvents that depend on two variables, the complex energy z and a perturbation parameter. The wave functions and the energies of the resonances are expanded in powers of. It is shown that the zero-order terms correspond to the standard CAP method and that higher-order corrections are significant. The introduction of a convergence operator allows to control the convergence of the perturbation series. Due to the discretization of the continuum, the series are generally asymptotic. Finally, we relate the perturbation series to numerically convenient Taylor series. The theory is illustrated on two model examples.

Proceedings of Symposia in Pure Mathematics, 2007

Quantum theory makes a sharp distinction between bound states and scattering states, the former associated with point spectrum and the latter with continuous spectrum. Resonances associated with quasi-stationary states bridge this distinction, and have posed mathematical challenges since the beginning of the Schrödinger theory. Here the development of the mathematical underpinnings of resonance theory in atomic physics is reviewed, with particular reference to the rôle of the (DC) Stark effect, and time-independent perturbations of bound states in the two-body problem in atomic and molecular physics.

American Journal of Physics, 2020

The second-order perturbative Stark effect on the ground state of hydrogen is a typical example presented in many standard texts on quantum mechanics. Some texts miss the fact that the scattering states are significant contributors to the perturbative energy correction. The inclusion of scattering states has wider applicability than to just the Stark effect. An explicit calculation involving a finite-square well with a perturbation is used to illustrate the importance of including scattering states into the calculation. The second-order correction to the ground-state energy is obtained in three distinct ways. The first involves altering the problem by imposing additional boundary conditions at large distances to make the positive-energy spectrum discrete. The second makes use of the continuum scattering states directly. The third bypasses the use of scattering states by solving a differential equation for the first-order correction to the wave function.

International Journal of Molecular Sciences, 2002

We present in this paper two new versions of Rayleigh-Schrödinger (RS) and the Brillouin-Wigner (BW) state-specific multi-reference perturbative theories (SS-MRPT) which stem from our state-specific multi-reference coupled-cluster formalism (SS-MRCC), developed with a complete active space (CAS). They are manifestly sizeextensive and are designed to avoid intruders. The combining coefficients c µ for the model functions φ µ are completely relaxed and are obtained by diagonalizing an effective operator in the model space, one root of which is the target eigenvalue of interest. By invoking suitable partitioning of the hamiltonian, very convenient perturbative versions of the formalism in both the RS and the BW forms are developed for the second order energy. The unperturbed hamiltonians for these theories can be chosen to be of both Mφller-Plesset (MP) and Epstein-Nesbet (EN) type. However, we choose the corresponding Fock operator f µ for each model function φ µ , whose diagonal elements are used to define the unperturbed hamiltonian in the MP partition. In the EN partition, we additionally include all the diagonal direct and exchange ladders. Our SS-MRPT thus utilizes a multi-partitioning strategy. Illustrative numerical applications are presented for potential energy surfaces (PES) of the ground ( 1 Σ + ) and the first delta ( 1 ∆) states of CH + which possess pronounced multi-reference character. Comparison of the results with the corresponding full CI values indicates the efficacy of our formalisms.

Chemical Physics, 2009

The retaining the excitation degree (RE) partitioning [R.F. Fink, Chem. Phys. Lett. 428 (2006) 461(20 September)] is reformulated and applied to multi-reference cases with complete active space (CAS) reference wave functions. The generalised van Vleck perturbation theory is employed to set up the perturbation equations. It is demonstrated that this leads to a consistent and well defined theory which fulfils all important criteria of a generally applicable ab initio method: The theory is proven numerically and analytically to be size-consistent and invariant with respect to unitary orbital transformations within the inactive, active and virtual orbital spaces. In contrast to most previously proposed multi-reference perturbation theories the necessary condition for a proper perturbation theory to fulfil the zeroth order perturbation equation is exactly satisfied with the RE partitioning itself without additional projectors on configurational spaces. The theory is applied to several excited states of the benchmark systems CH 2 , SiH 2 , and NH 2 , as well as to the lowest states of the carbon, nitrogen and oxygen atoms. In all cases comparisons are made with full configuration interaction results. The multi-reference (MR)-RE method is shown to provide very rapidly converging perturbation series. Energy differences between states of similar configurations converge even faster.